Thanks, Adam! I thought there should be some general method for solving P(n) = Q(K) where P, Q are quadratic polynomials over the integers, but didn't know where to go from there. * * * When there is an integer like 169 that belongs to two figurate numbers (H_7 = 169 = 13^2), I wonder if there is a set of 169 points in R^4 = R^2 x R^2 whose projection by p_12 onto one R^2 factor is the first figure and onto the other by p_34 is the second figure. I seem to recall asking this here some years ago for T_8 = 6^2, and that Michael Kleber showed that there's no set X of 36 points in R^4 with p_12(X) = triangle of side 8 and p_34(X) = square of side 6. Maybe there's a general statement about when such a thing can exist? —Dan —Dan Adam Goucher wrote: ----- Your equation is of the form: (quadratic in n) = (quadratic in K) ... ... -----